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Liouville theorem for entire functions


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1: 1.9 Calculus of a Complex Variable
Liouville’s Theorem
2: 15.16 Products
§15.16 Products
where A 0 = 1 and A s , s = 1 , 2 , , are defined by the generating function
Generalized Legendre’s Relation
For further results of this kind, and also series of products of hypergeometric functions, see Erdélyi et al. (1953a, §2.5.2).
3: 18.39 Applications in the Physical Sciences
Below we consider two potentials with analytically known eigenfunctions and eigenvalues where the spectrum is entirely point, or discrete, with all eigenfunctions being L 2 and forming a complete set. … The spectrum is entirely discrete as in §1.18(v). … The spectrum is entirely discrete as in §1.18(v). An important, and perhaps unexpected, feature of the EOP’s is now pointed out by noting that for 1D Schrödinger operators, or equivalent Sturm-Liouville ODEs, having discrete spectra with L 2 eigenfunctions vanishing at the end points, in this case ± see Simon (2005c, Theorem 3.3, p. 35), such eigenfunctions satisfy the Sturm oscillation theorem. …Both satisfy Sturm’s theorem. …
4: Errata
We now include Markov’s Theorem. … The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions. …
  • Chapter 1 Additions

    The following additions were made in Chapter 1:

  • Equation (25.2.4)

    The original constraint, s > 0 , was removed because, as stated after (25.2.1), ζ ( s ) is meromorphic with a simple pole at s = 1 , and therefore ζ ( s ) ( s 1 ) 1 is an entire function.

    Suggested by John Harper.

  • Section 27.20

    The entire Section was replaced.